We are now in a position to prove a pair of crucial properties for the subalgebra \({S}_{B}\) . First of all, we need to recall the notion of
pre-\(C^*\) -algebra according to [1]} or [2]} (where it is presented under the name of local-\(C^*\) -algebra). In a nutshell,
a pre-\(C^*\) -algebra
is a dense \(\ast \) -subalgebra of a \(C^*\) -algebra which is stable under holomorphic functional calculus. In the non-unital case (which is the relevant case for us) one has to restrict to holomorphic functions that satisfy \(f(0)=0\) .
